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Theorem sb4bOLD 2494
Description: Obsolete version of sb4b 2493 as of 21-Feb-2024. (Contributed by NM, 27-May-1997.) Revise df-sb 2063. (Revised by Wolf Lammen, 25-Jul-2023.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
sb4bOLD (¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑)))

Proof of Theorem sb4bOLD
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 nfeqf2 2388 . . . 4 (¬ ∀𝑥 𝑥 = 𝑡 → Ⅎ𝑥 𝑦 = 𝑡)
2 nfnf1 2151 . . . . . . 7 𝑥𝑥 𝑦 = 𝑡
3 id 22 . . . . . . 7 (Ⅎ𝑥 𝑦 = 𝑡 → Ⅎ𝑥 𝑦 = 𝑡)
42, 3nfan1 2192 . . . . . 6 𝑥(Ⅎ𝑥 𝑦 = 𝑡𝑦 = 𝑡)
5 equequ2 2026 . . . . . . . 8 (𝑦 = 𝑡 → (𝑥 = 𝑦𝑥 = 𝑡))
65imbi1d 344 . . . . . . 7 (𝑦 = 𝑡 → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑡𝜑)))
76adantl 484 . . . . . 6 ((Ⅎ𝑥 𝑦 = 𝑡𝑦 = 𝑡) → ((𝑥 = 𝑦𝜑) ↔ (𝑥 = 𝑡𝜑)))
84, 7albid 2216 . . . . 5 ((Ⅎ𝑥 𝑦 = 𝑡𝑦 = 𝑡) → (∀𝑥(𝑥 = 𝑦𝜑) ↔ ∀𝑥(𝑥 = 𝑡𝜑)))
98pm5.74da 802 . . . 4 (Ⅎ𝑥 𝑦 = 𝑡 → ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑))))
101, 9syl 17 . . 3 (¬ ∀𝑥 𝑥 = 𝑡 → ((𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ (𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑))))
1110albidv 1914 . 2 (¬ ∀𝑥 𝑥 = 𝑡 → (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑))))
12 df-sb 2063 . 2 ([𝑡 / 𝑥]𝜑 ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑦𝜑)))
13 ax6ev 1965 . . . 4 𝑦 𝑦 = 𝑡
1413a1bi 365 . . 3 (∀𝑥(𝑥 = 𝑡𝜑) ↔ (∃𝑦 𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑)))
15 19.23v 1936 . . 3 (∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑)) ↔ (∃𝑦 𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑)))
1614, 15bitr4i 280 . 2 (∀𝑥(𝑥 = 𝑡𝜑) ↔ ∀𝑦(𝑦 = 𝑡 → ∀𝑥(𝑥 = 𝑡𝜑)))
1711, 12, 163bitr4g 316 1 (¬ ∀𝑥 𝑥 = 𝑡 → ([𝑡 / 𝑥]𝜑 ↔ ∀𝑥(𝑥 = 𝑡𝜑)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 208  wa 398  wal 1528  wex 1773  wnf 1777  [wsb 2062
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-10 2138  ax-12 2169  ax-13 2383
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-ex 1774  df-nf 1778  df-sb 2063
This theorem is referenced by: (None)
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